# Centre of mass

The centre of mass is also known as the centre of gravity.

The centre of mass of a body has the property that the total moment of the object’s weight about any line through the centre of mass is zero.

If a body is freely suspended from a point on the body and hangs in stable equilibrium, it will hang so that the centre of mass is directly below the point of attachment.

In this sketch, a body is suspended from the point $A$, and its centre of mass $G$ lies directly below $A$.

For a planar shape (a lamina), the centre of mass is also the point on which it could be balanced horizontally.

If masses are given with a coordinate system, say there is a mass of $m_1$ at $(x_1,y_1)$, a mass of $m_2$ at $(x_2,y_2)$, …, and a mass of $m_n$ at $(x_n,y_n)$, then the moments property above can be used to calculate the centre of mass of the whole collection. It turns out that this centre of mass, $(\bar x,\bar y)$, has the property that $\bar x$ is the weighted mean of the $x_i$ (weighted by the masses $m_i$), and similarly for $\bar y$. So $\bar x=\frac{\sum_{i=1}^n m_ix_i}{\sum_{i=1}^n m_i}\qquad\text{and}\qquad \bar y=\frac{\sum_{i=1}^n m_iy_i}{\sum_{i=1}^n m_i}.$